COMP/MATH 553 Algorithmic Game Theory Lecture 2: Mechanism Design Basics. Sep 8, Yang Cai

Transcription

1 COMP/MATH 553 Algorithmic Game Theory Lecture 2: Mechanism Design Basics Sep 8, 2014 Yang Cai

2 An overview of the class Broad View: Mechanism Design and Auctions First Price Auction Second Price/Vickrey Auction Case Study: Sponsored Search Auction

3 [1] Broader View Mechanism Design (MD) Auction

4 ? What is It s the Science of Rule Mechanism Making. Design?

5 What is Mechanism Design? Understanding an existing game/economic system. Engineering part of Game Theory/Economics Existing System Predict Most of Game Theory/Economics devoted to Explain/predict the outcome. Outcome Mechanism Design reverse the direction Identifies the desired outcome/goal first! Achievable? Goal Asks whether the goals are achievable? If so, how? System Mechanism Design

6 Auctions Auctions Elections, fair division, etc. (will cover if time permits) MD Auction [1] Broader View

7 Auction example 1 Online Marketplace MD Auction [1] Broader View

8 Auction example 2 Sponsored Search MD Auction [1] Broader View

9 Auction example 3 Spectrum Auctions MD Auction [1] Broader View

10 SINGLE ITEM AUCTION

11 Single-item Auctions: Set-up Bidders Auctioneer v1 1 vi i Bidders: n vn have values on the item. These values are Private. Quasilinear utility: vi p, if wins. 0, if loses. Item

12 Auction Format: Sealed-Bid Auction Bidders v1 1 Auctioneer Item vi i n vn Sealed-Bid Auctions: 1. Each bidder i privately communicates a bid bi to the auctioneer in a sealed envelope, if you like. 2. The auctioneer decides who gets the good (if anyone). 3. The auctioneer decides on a selling price.

13 Auction Format: Sealed-Bid Auction Bidders v1 1 Auctioneer Item vi i n vn Sealed-Bid Auctions: Goal:1.Maximize social welfare. (Give itcommunicates to the bidder withathe value) Each bidder i privately bidhighest bi to the Natural Choice: Give it to the bidder with the highest bid. The only auctioneer selection rule wein use this lecture. a insealed envelope, if you like. 2. The auctioneer decides who gets the good (if anyone). 3. The auctioneer decides on a selling price.

14 Auction Format: Sealed-Bid Auction Bidders v1 1 Auctioneer Item vi i n vn Sealed-Bid Auctions: about athe 1. Each bidder i privatelyhow communicates bid bi to the selling price? auctioneer in a sealed envelope, if you like. 2. The auctioneer decides who gets the good (if anyone). 3. The auctioneer decides on a selling price.

15 Auction Format: Sealed-Bid Auction How about the selling price? Altruistic and charge nothing? Name the largest number you can... Fails terribly...

16 First Price Auction Pay you bid (First Price)? Hard to reason about. What did you guys bid? For two bidders,each bidding half of her value is a Nash eq. Why?

17 First Price Auction Game played last time Assume your value vi is sampled from U[0,1]. You won t overbid, so you will discount your value. Your strategy is a number di in [0,1] which specifies how much you want to discount your value, e.g. bi = (1 di) vi Game 1: What will you do if you are playing with only one student (picked random) from the class? Game 2: Will you change your strategy if you are playing with two other students? If yes, what will it be?

18 First Price Auction Pay you bid (First Price)? For two bidders,each bidding half of her value is a Nash eq. Why? How about three bidders? n bidders? Discounting a factor of 1/n is a Nash eq.

19 First Price Auction Pay you bid (First Price)? What if the values are not drawn from U[0,1], but from some arbitrary distribution F? bi(v) = E[maxj i vj vj v ] What if different bidders have their values drawn from different distributions? Eq. strategies could get really complicated...

20 First Price Auction Example [Kaplan and Zamir 11]: Bidder 1 s value is drawn from U[0,5], bidder 2 s value is drawn from U[6,7].

21 First Price Auction Example [Kaplan and Zamir 11]: Bidder 1 s value is drawn from U[0,5], bidder 2 s value is drawn from U[6,7]. o Nash eq. : bidder 1 bids 3 if his value is in [0,3], otherwise for b in (3, 13/3]:

22 First Price Auction Pay you bid (First Price)? Depends on the number of bidders. Depends on your information about other bidders. Optimal bidding strategy complicated! Nash eq. might not be reached. Winner might not value the item the most.

23 Second Price/Vickrey Auction Another idea Charge the winner the second highest bid. Seems arbitrary... But actually used in Ebay.

24 Second-Price/Vickrey Auction Lemma 1: In a second-price auction, every bidder has a dominant strategy: set its bid bi equal to its private valuation vi. That is, this strategy maximizes the utility of bidder i, no matter what the other bidders do. Super easy to participate in. (unlike first price) Proof: See the board.

25 Second-Price/Vickrey Auction Lemma 2: In a second-price auction, every truthful bidder is guaranteed non-negative utility. Proof: See the board.

26 Second Price/Vickrey Auction [Vickrey 61 ] The Vickrey auction is has three quite different and desirable properties: (1) [strong incentive guarantees] It is dominant-strategy incentivecompatible (DSIC), i.e., Lemma 1 and 2 hold. (2) [strong performance guarantees] If bidders report truthfully, then the auction maximizes the social welfare Σi vixi, where xi is 1 if i wins and 0 if i loses. (3) [computational efficiency] The auction can be implemented in polynomial (indeed linear) time.

27 What s next? These three properties are criteria for a good auction: More complicated allocation problem Optimize Revenue

28 Case Study: Sponsored Search Auction

29 Sponsored Search Auction

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31 Sponsored Search Auctions: Set-up Bidders (advertisers) Slots v1 1 vn k slots for sale. Slot j has click-through-rate (CTR) αj. Bidder i s value for slot j is αjvi. Two complications: o Multiple items o Items are non identical 1 αj j n α1 vi i Auctioneer/ Google αk k

32 Sponsored Search Auction: Goal (1)DSIC. That is, truthful bidding should be a dominant strategy, and never leads to negative utility. (2) Social welfare maximization. That is, the assignment of bidders to slots should maximize Σvixi, where xi now denotes the CTR of the slot to which i is assigned (or 0 if i is not assigned to a slot). Each slot can only be assigned to one bidder, and each bidder gets only one slot. (3) Polynomial running time. Remember zillions of these auctions need to be run every day!